Final answer:
The distribution of the sample mean is approximately normal due to the Central Limit Theorem.
This correct answer is (a)
Step-by-step explanation:
The shape of the distribution of the sample mean is approximately normal.
This phenomenon can be attributed to the Central Limit Theorem (CLT), which states that if you draw samples of a sufficient size from a population, the distribution of sample means will tend to be normal.
This holds true regardless of whether the population distribution itself is normal or not.
When the sample size is large enough, usually n ≥ 30, the sampling distribution of the sample mean will approach normality. As samples get larger, they better reflect the population's characteristics.
Thus, even for a population with a skewed or non-normal distribution, the distribution of the sample mean will approximate a normal distribution, given that the sample size is sufficiently large.
It's essential to note that the mean of the sample means will equal the population mean, and the standard error of the mean, which is the standard deviation of the sample means, will equal the population standard deviation divided by the square root of the sample size (n).
This correct answer is (a)
Your correct question is: (a) What is the shape of the distribution of the sample mean? Why? A. The distribution is approximately normal because of the Central Limit Theorem. B. The distribution is approximately normal because of the Law of Large Numbers. C. The distribution is skewed left because the right tail is longer than the left tail. D. The distribution is skewed right because the right tail is longer than the left tail. (b) Compute a 95% confidence interval for the mean number of children of all couples who have been married 7 years. Interpret this interval. that who have been married for 7 years have a mean number of children between and (Type integers or decimals rounded to two decimal places as needed. Use ascending order.) (c) Compute a 99% confidence interval for the mean number of children of all couples who have been married 7 years. Interpret this interval. that who have been married for 7 years have a mean number of children between and (Type integers or decimals rounded to two decimal places as needed. Use ascending order.)